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A152149
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Decimal expansion of the angle B in the doubly golden triangle ABC.
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11
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6, 5, 7, 4, 0, 5, 4, 8, 2, 9, 7, 6, 5, 3, 2, 5, 9, 2, 3, 8, 0, 9, 6, 8, 5, 4, 1, 5, 2, 9, 3, 9, 7, 1, 2, 6, 5, 4, 1, 4, 9, 5, 9, 4, 6, 4, 8, 7, 8, 3, 9, 3, 7, 0, 7, 8, 2, 0, 9, 2, 8, 0, 8, 5, 8, 8, 5, 3, 9, 5, 0, 6, 1, 3, 8, 1, 7, 7, 3, 5, 0, 7, 0, 1, 7, 1, 5, 1, 6, 5, 4, 4, 0, 5, 2, 2, 7, 8, 0, 5, 2, 8, 1, 2, 6
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OFFSET
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0,1
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COMMENTS
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There is a unique (shape of) triangle ABC that is both side-golden and angle-golden. Its angles are B, C=t*B and A=pi-B-t*B, where t is the golden ratio. "Angle-golden" and "side-golden" refer to partitionings of ABC, each in a manner that matches the continued fraction [1,1,1,...] of t. (The partitionings are analogous to the partitioning of the golden rectangle into squares by the removal of exactly 1 square at each stage.)
For doubly silver and doubly e-ratio triangles, see A188543 and A188544.
For the side partitioning and angle partitioning (i,e, constructions) which match arbitrary continued fractions (of sidelength ratios and angle ratios), see the 2007 reference.
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REFERENCES
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Clark Kimberling, "A new kind of golden triangle," in Applications of Fibonacci Numbers, Proc. Fourth International Conference on Fibonacci Numbers and Their Applications, Kluwer, 1991.
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LINKS
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FORMULA
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B is the number in [0,Pi] such that sin(B*t^2)=t*sin(B), where t=(1+5^(1/2))/2, the golden ratio.
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EXAMPLE
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The number B begins with 0.65740548 (equivalent to 37.666559... degrees).
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MATHEMATICA
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r = (1 + 5^(1/2))/2; Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]
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PROG
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(PARI) t=(1+5^(1/2))/2; solve(b=.6, .7, sin(b*t^2)-t*sin(b)) \\ Iain Fox, Feb 11 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Keyword:cons added and offset corrected by R. J. Mathar, Jun 18 2009
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STATUS
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approved
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